The poll shows a move toward blaming Obama following a similar survey conducted earlier in the month.

"The number who would hold congressional Republicans responsible has gone down by 5 points since early September, and the number who would blame Obama is up 3 points in that same time," CNN Polling Director Keating Holland pointed out. "Those changes came among most demographic groups."

The poll’s findings come on the heels of the GOP-led House approving a plan to fund the government, but it also delays provisions of the Affordable Care Act for one year. Republicans say the deal stops a shutdown and prevents Obamacare from kicking in, something they say their constituents are demanding.

"A majority of Republicans think that blocking Obamacare is more important than approving a budget agreement," said Holland. "So do tea party supporters, regardless of their partisan affiliation."

A leader of the tea party Republicans, Sen. Ted Cruz, of Texas insisted Senate Democrats would be to blame if the shutdown occurs.

"The House has twice now voted to keep the government open. And if we have a shutdown, it will only be because when the Senate comes back, Harry Reid says, `I refuse even to talk,'" said Cruz.

House Republicans gathered on the Capitol Steps on Sunday, hours after passing their latest funding bill, calling the Senate out for not working on the weekend.

"If they really cared whether this government would shut down or not, they would be here," Rep. Tim Griffin of Arkansas said. "This is the old football strategy. When you get to where you want to be in a football game, you run out the clock," the Washington Times reported.

The Democrat-led Senate is set to reconvene Monday afternoon, just 10 hours before the shutdown deadline.

Most of the public says they’re tired of the political squabble, with 60 percent of people saying it is more important to avoid a government shutdown than it is to change the healthcare law, according to the CNN poll.

Other poll findings:

• When it comes to Obamacare, 57 percent of people oppose it, 38 percent support it.
• Four in 10 people oppose Obamacare because they believe it’s too liberal.
• About 68 percent of people say a government shutdown would hurt the country.